A redditor named Erebor has shared some proof and a way to make synths constantly work while dead. Although I think some of you know this as common knowledge but for those who doesn't know, then you should read on.
The same goes for using a synth whilst dead and your team wiping before you are revived as shown in the video below.
So those videos are proof synths won't work after you are dead (or looking at a wipe screen) for some time after popping the synth. Here comes proof IT DOES WORK if you pop them right before coming back to life!
In another video (which you can watch below), you see Erebor wiping on the Crota's end raid, quickly going into menu's and self-rezzing before the darkness timer runs out. He then was revived back alive with heavy ammo. The truth is though it doesn't have to be that fast.
Here's yet another video of him dying on a KF NM Golgoroth fight, waiting until there is less than 10 seconds on the revive timer, popping the synth and getting his ammo!
So, with all these proof, I can safely say that you can use synths while you're dead. But if you still don't believe it after all these stuff, then you better try it for yourself.
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Credits: Special thanks to Erebor for sharing this very helpful tip.